Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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PERSISTENCE OF PERIODIC TRAJECTORIES OF PLANAR SYSTEMS UNDER TWO PARAMETRIC PERTURBATIONS

  • Published : 2007.05.31
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Abstract

We consider a two parametric family of the planar systems with the form $\dot{x}=P(x,\;y)+{\in}_1p_1(x,\;y)+{\in}_2p_2(x,\;y)$, $\dot{y}=Q(x,\;y)+{\in}_1p_1(x,\;y)+{\in}_2p_2(x,\;y)$, where the unperturbed equation(${\in}_1={\in}_2=0$) is assumed to have at least one periodic solution or limit cycle. Our aim here is to study the behavior of the system under two parametric perturbations; in fact, using the Poincare-Andronov technique, we impose conditions on the system which guarantee persistence of the periodic trajectories. At the end, we apply the result on the Van der Pol equation ; where, we consider the effect of nonlinear damping on the equation. Also the Hopf bifurcation for the Van der Pol equation will be investigated.

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References

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  2. EXTENSION OF CHICONE'S METHOD FOR PERTURBATION SYSTEMS OF THREE PARAMETERS WITH APPLICATION TO THE LIENARD SYSTEM vol.22, pp.03, 2012, https://doi.org/10.1142/S0218127412500654